On approximation of BSDE and multi-step MLE-processes

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and Quantitative Risk

R E S E A R C H Open Access

On approximation of BSDE and multi-step

MLE-processes

Yu A. Kutoyants

Received: 2 January 2016 / Accepted: 15 June 2016 /

© The Author(s). 2016Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Abstract We consider the problem of approximation of the solution of the backward

stochastic differential equations in Markovian case. We suppose that the forward equation depends on some unknown finite-dimensional parameter. This approxi-mation is based on the solution of the partial differential equations and multi-step estimator-processes of the unknown parameter. As the model of observations of the forward equation we take a diffusion process with small volatility. First we establish a lower bound on the errors of all approximations and then we propose an approx-imation which is asymptotically efficient in the sense of this bound. The obtained results are illustrated on the example of the Black and Scholes model.

Keywords Backward stochastic differential equation·Parameter estimation·

Multi-step MLE-process·Small noise·Black and Scholes model

MSC 2000 Classification 62M05

Introduction

We consider the problem of approximation of the solution of the backward stochastic differential equation (BSDE) in the so-called Markovian case. Let us recall some basics of BSDEs. We are given a stochastic differential equation (calledforward)

dXt =S(t, Xt)dt+σ (t, Xt)dWt, X0=x0, 0≤t ≤T ,

Y.A. Kutoyants ()

Laboratoire Manceau des Math´ematiques, Universit´e du Maine Le Mans, France and National Research University “MPEI”, Moscow, Russia

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where S (t, x) is the drift coefficient, σ (t, x)2 is the diffusion coefficient, and Wt,0 ≤ t ≤ T is a standard Wiener process. In addition, we have two

func-tions f (t, x, y, z) and (x) and we must construct such couple of processes (Yt, Zt,0≤t ≤T )that the solution of the equation

dYt = −f (t, Xt, Yt, Zt)dt+Zt dWt, Y0, 0≤t≤T ,

(calledbackward) has the final valueYT = (XT). Such BSDEs were first

intro-duced by Bismut in 1973 (Bismut 1973) in the linear case and the general theory was developed by Pardoux and Peng (1990) (Pardoux and Peng 1990). The Marko-vian caseconsidered in this work was studied by Pardoux and Peng (Pardoux and Peng 1992), see Section 4 in El Karoui et al. (1997) as well. This model is also called forward-backward stochastic differential equation(FBSDE) (El Karoui et al. 1997). The construction of the backward equation is realized as follows. Suppose that u (t, x)satisfies the parabolic partial differential equation

∂u

∂t +S (t, x) ∂u

∂x +

1 2σ (t, x)

2∂2u

∂x2 = −f

t, x, u, a (t, x)∂u ∂x

,

with the final condition u (T , x) = (x). Let us let Yt = u (t, Xt) , andZt =

σ (t, Xt) ux(t, Xt). Then, by Itˆo’s formula

dYt =

∂u

∂t (t, Xt)+S (t, Xt) ∂u

∂x(t, Xt)+ 1

2σ (t, x)2 ∂2u ∂x2(t, Xt)

dt

+σ (t, Xt)

∂u

∂x(t, Xt)dWt

= −f (t, Xt, Yt, Zt)dt+ZtdWt, Y0=u (0, X0) , 0≤t ≤T . The final valueYT = u (T , XT) = (XT). Therefore, if we have the solution

u (t, x), then we immediately obtain the BSDE.

We are interested by the problem of approximation of(Yt, Zt,0≤t ≤T )in the

situation, where the forward equation contains some unknown finite-dimensional parameterϑ:

dXt =S(ϑ, t, Xt)dt+σ (ϑ, t, Xt)dWt, X0=x0, 0≤t≤T .

Then the solution of the PDE u = u (t, x, ϑ ). We cannot simply let Yt =

u(t, Xt, ϑ )because we do not knowϑ. Of course, thenaturalway to approximate

Yt andZt is to estimate first the unknown parameterϑ with the help of some

esti-matorϑ¯ and then to put, say, Y¯t = u(t, Xt,ϑ)¯ . We can guess that ifϑ¯ is a good

estimator ofϑ, thenY¯t will be a good estimator ofYt. There are several problems,

that are interesting to study in this framework. We must understand what the condi-tions imposed on the estimatorϑ¯ that allow us to say that it is good. We consider that a good estimator has the following properties.

1. To estimateYt we need an estimator, which is constructed by the first

obser-vations of the solution of the forward equation up to time t, i.e., ϑ¯t = ¯

ϑt(Xs,0≤s≤t), 0< t ≤T.

2. As we need such estimator for allt ∈(0, T]we suppose that its calculation must

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3. The error of estimation, say,Eϑ₀ϑ¯t−ϑ02must be as minimal as possible.

Thereforeϑ¯ is an estimator-processϑ¯ = ϑ¯t,0< t ≤T

. Of course, the con-struction of such estimator-process is an intermediate problem. The main problem is to obtain a good approximations ofYt andZt. In particular, we must show that the

approximations

¯

Yt =u(t, Xt,ϑ¯t), Z¯t =ux(t, Xt,ϑ¯t) σ (ϑ¯t, t, Xt)

are in some sense asymptotically optimal, i.e., it is impossible to have approximations of these processes with asymptotic errors smaller than that ofY¯tandZ¯t.

The goal of the study initiated in Kutoyants and Zhou (2014) is to realize such a program for three models of observations of the forward equation. As is usual in statistics, we consider situations where it is possible to have a consistent estimation of the unknown parameters and processes. Therefore, we are interested by the following well-known models of observations.

• Diffusion process with an unknown parameter in the drift coefficient andsmall noiseorsmall volatility

dXt =S (ϑ, t, Xt)dt+εσ (t, Xt)dWt, x0, 0≤t ≤T . (1) Here the timeTof observationsXT =(Xt,0≤t ≤T )is fixed and the limit

corresponds toε→0.

• Diffusion process

dXt =S (t, Xt)dt+σ (ϑ, t, Xt)dWt, X0, 0≤t ≤T , (2)

observed in the discrete times Xn = Xt0, Xt1, . . . Xtn

, ti = iT_n. Here the

unknown parameter is in the volatility coefficient and the limit corresponds to n→ ∞(high frequency model of observations). The timeT of observations is fixed.

• Ergodic diffusion process

dXt =S (ϑ, Xt)dt+σ (Xt)dWt, X0, 0≤t≤T . (3)

Here the unknown parameterϑis in the drift coefficient, we have continuous time observationsXT =(Xt,0≤t ≤T )and the limit isT → ∞.

Of course there are other possible statements. For example, it can be considered themixtureof discrete time and ergodic diffusion. This corresponds to the equation

dXt =S (ϑ, Xt)dt+σ (ϑ, Xt)dWt, X0, 0≤t≤Tn

and observationsXn=Xt0, Xt1, . . . Xtn

. Here maxi|ti−ti−1| →0 andTn→ ∞.

Such a model of parameter estimation was studied, e.g., in Kamatani and Uchida (2015) and Uchida and Yoshida (2014). It is possible to consider the mixture of discrete-time and small noise models, to consider the model withXt → ±∞ or

the models with null recurrent forward equation etc. It will be interesting to see the statements of the statistical problems in non-Markovian cases for more general models.

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ϑ_t,0 < t ≤ T such thatY_t =u(t, Xt, ϑt∗)→ Yt and the error of approximation EϑYt−Yt

2_{is asymptotically minimal. In the earlier works Kutoyants and Zhou}

(2014) and Gasparyan and Kutoyants (2015); (Abakirova, A and Kutoyants, YA: On approximation of the BSDE. Large samples approach.In preparation) (see the review of these works in Kutoyants (2014)) we considered the approximation of the solution of BSDEs with a learning interval of fixed length.

The optimality of estimators ofYt andZt is understoud as follows. We define for

each model a normalization functionϕ → 0, i.e.,ϕε → 0 asε → 0,ϕn → 0 as

n→ ∞, andϕT →0 asT → ∞.

We propose the lower bounds on the risks of all estimators

lim

δ→0

lim

ε,n,T

sup

|ϑ−ϑ0|<δ

EϑY¯t−Yt

ϕ

2≥D (ϑ0)2,

which allow us to define theasymptotically efficient estimatorsY_tofYt as follows

lim

δ→0ε,n,Tlim _|_ϑ₋sup_ϑ₀_|_<δEϑ

Yt−Yt

ϕ

2=D (ϑ0)2.

We suppose that the last equality takes place for allϑ0∈and allt∈(0, T]. We also have a similar bound in the problem of estimation ofZt. For models (1) and (3)

these bounds are slight modifications of the Hajek-Le Cam lower bound (Ibragimov and Has’minskii 1981) and for model (2) the lower bound is similar to Jeganathan’s lower bound (Jeganathan 1983).

We take the quadratic loss function just for simplicity of exposition. For all men-tioned models, the similar lower bounds and corresponding estimator processes can be proved for more general loss functions.

The approximation of the solution of BSDEs in the Markovian case were initiated in the work Kutoyants and Zhou (2014), where the model of small volatility was considered. The parameterϑwas supposed one-dimensional and the approximation-processYt,ε was defined fort ∈[τ, T], whereτ >0 is a fixed value.

In the work Gasparyan and Kutoyants (2015), we considered the model of discrete-time observations (2) and the one-step MLE-process which allowed us to construct an estimator-processYtk,nfor the valuestk,n∈[τ, T], whereτ >0 is fixed.

The case of ergodic diffusion process is considered in the work (Abakirova, A and Kutoyants, YA: On approximation of the BSDE. Large samples approach.In preparation), which is still in progress.

The main contribution of the present work is due to a new class of estimator-processes called multi-step MLE-process introduced in Kutoyants (2015). These estimator-processes allow us to construct the approximations of the solutions of BSDEs for three above mentioned models with vanishing learning intervals (models (1) and (2)) or negligible with respect to the whole volume of observations learning interval (model (3)). Here we consider the model (1) only. The models (2) and (3) we leave to study later.

In the present work, we consider the small volatility model where we suppose that the unknown parameter is multi-dimensional and the approximation processYt,ε we

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caseτε = εδ → 0 and, moreover, to chooseδ close to 2. The relations between

the choice ofδ and the multi-step MLE-processes are the following. Ifδ ∈ (0,1), then we use the one-step MLE-processϑt,ε ; ifδ ∈ [1,43), then we use the two-step

MLE-processϑt,ε; ifδ∈ [43,32), then we use the three-step MLE-processϑt,ε.

In the work Kutoyants (2015) we aleady studied the multi-step MLE-process for ergodic diffusion process, and the structure of estimator-process proposed in the present work is quite similar.

Note that the multi-step, like the well-known one-step ML-estimators, are based on the so-calledFisher-scoredevice proposed by Fisher in 1925 (Fisher 1925) and studied by Le Cam in 1956 (Le Cam 1956). Let us recall this construction. Suppose that we haveni.i.d. r.v.’sXn=(X1, . . . , Xn)with smooth density functionf (ϑ, x)

and denote (ϑ, x)=lnf (ϑ, x). The maximum likelihood equation is

n

j=1 ˙

ϑˆn, Xj

=0.

Here and in the rest of the paper dot means derivation w.r.t.ϑ. If we expand it at the vicinity of the true valueϑ0, we obtain

n

j=1 ˙

ϑ0, Xj+

ˆ

ϑn−ϑ0

n

j=1 ¨

˜

ϑn, Xj

=0.

Therefore

ˆ

ϑn=ϑ0−

n j=1˙

ϑ0, Xj

n

j=1¨

˜

ϑn, Xj

=ϑ0+√1

n

1

√

n

n j=1˙

ϑ0, Xj

−1 n

n j=1¨

˜

ϑn, Xj

. (4)

Note that

−1

n

j₌1 ¨

ϑ0, Xj

−→I(ϑ0)=

˙

(ϑ0, x)2f (ϑ0, x)dx, (5)

whereI(ϑ0)is the Fisher information.

Suppose that we have a preliminary estimatorϑ¯nsuch that √

nϑ¯n−ϑ0=⇒N(0, D (ϑ0)) , D (ϑ0) >I(ϑ0)−1.

Keeping in mind the relations (4)-(5), the one-step MLEϑnis defined as follows

ϑ_n= ¯ϑn+

1

√

n

_¯ ϑn, Xn

Iϑ¯n

, nϑ, Xn=

1

√

n

j=1 ˙

ϑ, Xj.

This estimator is already asymptotically efficient because its limit variance is I(ϑ0)−1:

√

nϑn−ϑ0

=⇒N0,I(ϑ0)−1.

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Moreover, this device can be applied even in the case of preliminary estimator with the rate of convergence worse than√n(see, e.g., Robinson (1988) and Kamatani and Uchida (2015)). For continuous-time stochastic processes such a construction was used, for example, in Skorohod and Khasminskii (1996).

The one-step MLE-process, introduced in Kutoyants (2015), for this model of observations can be written as follows. Let us denoteϑ¯N the premilinary estimator

constructed by the firstN =nδobservationsXN =(X1, . . . , XN)withδ∈(1₂,1).

Then the one-step MLE-processϑn =

ϑ_k,n , N+1≤k≤n

is defined by the equality

ϑ_k,n = ¯ϑN+I

_¯ ϑN

−11

k

j=N+1 ˙

ϑ¯N, Xj

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and fork= [sn], s∈(0,1]we have the convergence

√

kϑ_k,n −ϑ0=⇒N

0,I(ϑ0)−1

.

Heresis fixed andn→ ∞. Thereforeϑ_n is agood estimator, i.e.,ϑ_k,n depends onXk =(X1, . . . , Xk), is easy to calculate and is asymptotically efficient because it

is asymptotically equivalent to the MLE. For the details see Kutoyants and Motrunich (2016).

The one-step MLE-process in the case of ergodic diffusion forward Eq. 3 can be illustrated as follows. Suppose that we have a preliminary estimatorϑ¯Tδconstructed

by the observationsXTδ = Xt,0≤t≤Tδ

with δ ∈ (1₂,1]. Then the one-step MLE-processϑ_t,T , Tδ < t ≤ T based on the Fisher-score device (4), (6) has the following form

ϑt,T = ¯ϑTδ+Iϑ¯_Tδ−1 t

Tδ ˙

Sϑ¯_Tδ, Xs

t σ (Xs)2

dXs−Sϑ¯Tδ, Xsds. (7)

This estimator-process is asymptotically efficient (t=rT;r∈(0,1])

√

tϑ_t,T −ϑ0=⇒N

0,I(ϑ0)−1

(see Kutoyants (2015)) and provides asymptotically efficient estimator-processes Yt,T =u

t, Xt, ϑt,T

, Zt,T =ux

t, Xt, ϑt,T

σ (Xt) (8)

of the solution(Yt, Zt)of the BSDE.

Forward equation with small volatility

We are given the functionf (t, x, y, z)defined on [0, T]×Rk×R×Rk, function (x), x∈Rkandk-dimensional diffusion process (forward)

dXt =S (ϑ, t, Xt)dt+εσ (t, Xt)dWt, X0, 0≤t ≤T . (9)

Hereϑ∈⊂Rd,is an open bounded set andWt =

Wt1, . . . , Wtk

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Introduce the conditionL.

The functionsf (t, x, y, z) , (x),vectorS (ϑ, t, x)=(Sl(ϑ, t, x) , l=1, . . . , k)

andk×kmatrixσ (t, x)=(σlm(t, x))are smooth

|S (ϑ, t, x)−S (ϑ, t, y)| + |σ (t, x)−σ (t, y)| ≤L|x−y|,

|f (t, x, y1, z1)−f (t, x, y2, z2)| ≤C[|y1−y2| + |z1−z2|]

and satisfy(p >0)

|S (ϑ, t, x)| + |σ (t, x)| ≤C (1+ |x|) ,

|f (t, x, y, z)| + |(x)| ≤C1+ |x|p.

We must find a couple of stochastic processes X_t,ε, Z_t,ε ,0≤t≤T which approximate well the solution of the BSDE

dYt = −f (t, Xt, Yt, Zt)dt+ZtdWt, Y0, 0≤t ≤T (10)

satisfying the conditionYT = (XT).

Let us denotext(ϑ) =

xt(1)(ϑ), . . . , x (k) t (ϑ )

,0 ≤ t ≤ T the solution of the system of ordinary differential equations

dxt(ϑ )

dt =S (ϑ, t, xt(ϑ )) , x0, 0≤t ≤T .

The true value is ϑ0 and we let xt = xt(ϑ0). We have the estimates: with

probability 1

sup

0≤t≤T|

Xt−xt| ≤Cε sup 0≤t≤T|

Wt| (11)

and for anyp >0

sup

0≤t≤T

Eϑ₀|Xt−xt|p≤Cεp. (12)

For the proof see, e.g., Kutoyants (1994).

We have a family of problems of parameter estimation by observations Xt = (Xs,0≤s≤t), wheret ∈ (0, T] and therefore we need a family of estimators

¯

ϑt,ε,0< t ≤ T. Let

Ck₍_[0_{, t}_]_{) ,}_B t

be a measurable space of continuous vector-functions on [0, t] with Borelianσ-algebraBt. Denote by

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Note that these measures are equivalent (see Liptser and Shiryaev (2001)) and the likelihood ratio function is

Lϑ, Xt = dP

(ε,t ) ϑ

dP(ε,t )₀

Xt=exp ₁

ε2 t

0 S (ϑ, s, Xs)A(s, Xs)

−1_d_X t

−₂1

ε2 t

0S (ϑ, s, Xs)A(s, Xs)

−1_{S (ϑ, s, X} s)ds

, ϑ∈.

Here P(ε,t )₀ is the measure, which corresponds to the observations (9) with

S (ϑ, t, Xt)≡0. The matrixA(s, x)is

Alm(s, x)=

σ (s, x)∗σ (s, x)_lm, l, m=1, . . . , k. Recall that the MLEϑˆε,t is defined by the equation

Lϑˆε,t, Xt

= sup

ϑ∈

Lϑ, Xt. (13)

Introduce theRegularity conditionsR.

1. The functionS (ϑ, t, x)is two-times continuously differentiable w.r.t.ϑand the

derivatives are Lipschitz in x.

2. We suppose that there exists a positive constant m such that for any realλ∈Rk

we have

m−1λ2≤λ∗A(s, x) λ≤mλ2. (14)

3. The Fisher information matrix

It(ϑ )=

t

0 ˙

S(ϑ, s, xs(ϑ ))∗A(s, xs(ϑ ))−1S˙(ϑ, s, xs(ϑ ))ds,

is uniformly nondegenerate: inf

ϑ∈|λinf|=1λ

∗_I

t(ϑ )λ >0

Hereλ∈Rddot means derivation w.r.t.ϑandS˙(ϑ, s, x)isk×d-matrix. 4. Identifiability condition : for anyν >0and anyt ∈(0, T]the estimate

inf

ϑ0∈|ϑ−infϑ0|>ν

t

0 δ (s, xs, ϑ, ϑ0)

∗_A_{(s, x}

s)−1δ (s, xs, ϑ, ϑ0)ds >0

holds. Hereδ (s, xs, ϑ, ϑ0)=S (ϑ, s, xs)−S (ϑ0, s, xs).

The Regularity conditions allow us to prove the folowing properties of the MLE

ˆ

ϑε,t, t∈(0, T].

1. It is uniformly consistent: for anyν >0and any compactK⊂ lim

ε→0ϑsup0∈K

P(ε,t )_ϑ₀ ϑˆε,t−ϑ0> ν

=0.

2. Uniformly on compactsK⊂asymptotically normal ε−1ϑˆε,t−ϑ0

=⇒N0,It(ϑ0)−1

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3. The polynomial moments converge and it is asymptotically efficient.

These properties were established in Kutoyants (1994) in the case of the one-dimensional diffusion processes (9). There is no essential dificulties to apply the same proof in our case. The presented below multi-step MLE-processes have exactly the same asymptotic properties, but can be calculated more easily.

Introduce the family of functions

U =u(t, x, ϑ ), t∈[0, T], x∈Rk, ϑ ∈

such that for allϑ∈the functionu(t, x, ϑ )satisfies the PDE

∂u

∂t +

k

l=1

Sl(ϑ, t, x)

∂u ∂xl +

ε2 2

k

l,m=1

Al,m(t, x)

∂2u ∂xl∂xm

= −f

t, x, u, ε

k

l=1

σlm(t, x)

∂u ∂xm

and the conditionu(T , x, ϑ )=(x), x∈Rk.

The limit of the function u (t, x, ϑ )as ε → 0 we denote as u_◦(t, x, ϑ ). The functionu_◦(t, x, ϑ )satisfies the equation

∂u_◦ ∂t +

k

l=1Sl(ϑ, t, x) ∂u_◦ ∂xl = −f

t, x, u◦, ε

k

l=1σlm(t, x) ∂u_◦ ∂xm

with the final valueu_◦(T , x, ϑ)=(x). Belowu˙_◦(t, x, ϑ )andu¨_◦(t, x, ϑ )means the derivative of this function w.r.t.ϑ.

Introduce the conditionU

1. The functionu (t, x, ϑ )is two-times continuously differentiable w.r.t.ϑand the

derivativesu (t, x, ϑ )˙ andu (t, x, ϑ )¨ are Lipschitz w.r.t. x unifoormly inϑ∈. 2. The functionu (t, x, ϑ )and its derivativesu (t, x, ϑ )˙ andu˙x(t, x, ϑ )converge

uniformly int ∈[0, T]tou◦(t, x, ϑ ) ,u˙◦(t, x, ϑ ) ,u˙_◦,x(t, x, ϑ )respectively.

The sufficient conditions providing these properties ofu (t, x, ϑ )can be found in Freidlin and Wentzell (1998), Theorem 2.3.1. Note that the derivativesu (t, x, ϑ )_˙ and

¨

u (t, x, ϑ )satisfy the linear PDE of the same type.

If we letYt =u (t, Xt, ϑ ), then by Itˆo’s formula we obtain BSDE (10) with

Zt =

Z1t, . . . , Zkt

, Ztm=ε

k

l=1

σml(t, Xt) uxl(t, Xt, ϑ ) .

Recall that our goal is to construct an asymptotically efficient approximation of the couple(Yt, Zt). To compare all possible estimators we introduce the lower bounds on

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Theorem 1Suppose that the conditions L,R andUare fulfilled. Then for all estimatorsY¯t andZ¯t and allt ∈(0, T]we have the relations

lim

ν_→0εlim_→0|ϑ−supϑ0|≤ν

ε−2EϑY¯t−Yt 2

≥ ˙u_◦(t, xt, ϑ0)∗It(ϑ0)−1u˙0(t, xt, ϑ0) , (15)

lim

ν_→0εlim_→0|ϑ−supϑ0|≤ν

ε−4EϑZ¯t −Zt 2

≥(u˙_◦)x(t, xt, ϑ0)∗It(ϑ0)−

1

2_{σ (t, x}_t₎2_. (16)

ProofWe first verify that the family of measures is locally asymptotically normal (LAN) and then we apply the proof of the Hajek-Le Cam lower bound (Ibragimov and Has’minskii 1981), which provides us (15), (16). We present here the necessary modification of the proof given in Ibragimov and Has’minskii (1981). Usually this inequality is considered for the risk likeEϑϑ¯ε−ϑ

2

and we are interested by the riskEϑY¯t,ε−Yt2, whereYtis a random process. Another point, the random vector

t(see below), in general, is asymptotically normal and, in our case, it has Gaussian

distribution, that is why the proof is slightly simplified.

Let us denoteϕε = εIt−

1

2 where I_t = I_t_(ϑ₀₎ and introduce the normalized likelihood ratio

Zt,ε(v)=

Lϑ0+ϕεv, Xt

L (ϑ0, Xt) , v∈Vε= {v:ϑ0+ϕεv∈}.

We can write

lnZt,ε(v) =

1 ε

t

0 [S (ϑ0+ϕεv, s, Xs)−S (ϑ0, s, Xs)]σ (s, Xs)

−1_d_W s

−₂1

ε2 t

0

[S (ϑ0+ϕεv, s, Xs)−S (ϑ0, s, Xs)]∗σ (s, Xs)−1 2

ds

= v∗t−

1

2|v|2+rε,

whererε→0 and the vector

t =It(ϑ0)−

1 2

t

0 ˙

S(ϑ0, s, xs) σ (s, xs)−1dWs ∼N(0,J) .

HereJis a unitd×dmatrix.

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Below,M >0,KM is a cube inRdwhose vertices have coordinates±M, so that its volume is(2M)dandϑv=ϑ0+ϕεv.

sup

|ϑ−ϑ0|≤ν

EϑY¯t,ε−Yt2= sup

|ϑ−ϑ0|≤ν

EϑY¯t−u (t, Xt, ϑ )2

= sup

|ϕεv|≤ν

Eϑ_vY¯t −u (t, Xt, ϑ0+ϕεv)2

≥ sup

v∈KM

Eϑ_vY¯t−u (t, Xt, ϑ0+ϕεv)2

≥ 1

(2M)d

KM

EϑvY¯t−u (t, Xt, ϑ0+ϕεv) 2

dv

= 1

(2M)d

KM

Eϑ0Zt,ε(v)Y¯t−u (t, Xt, ϑ0+ϕεv) 2

dv.

We have

u (t, Xt, ϑ0+ϕεv) = u (t, Xt, ϑ0)+ ˙u (t, Xt, ϑ0)∗ϕεv

+ε 1

0 [u (t, X˙ t, ϑ0+rϕεv)− ˙u (t, Xt, ϑ0)]

∗_I−12

t vdr = u (t, Xt, ϑ0)+ ˙u (t, Xt, ϑ0)∗ϕεv+ϕεhε,

where|hε| ≤Cε. Hence, if we denote

¯

bε =ε−1Y¯t−u (t, Xt, ϑ0), u˙ = ˙u (t, Xt, ϑ0)∗I− 1 2

t

and introduce such vectorv¯εthatb¯ε= ˙u (t, Xt, ϑ0)∗I− 1 2

t v¯ε, then we can write

ε−2Eϑ₀Zt,ε(v)Y¯t−u (t, Xt, ϑ0+ϕεv)2 =Eϑ0Zt,ε(v)

b¯t− ˙u (t, Xt, ϑ0)∗I− 1 2

t v

2(1+O(ε))

=Eϑ₀Zt,ε(v)u˙∗(v¯ε−v)2(1+O(ε)) .

Further, we use the following result known as Scheff´e’s lemma

Lemma 1Let the random variablesZε≥0, ε∈(0,1]converge in probability to

the random variableZ≥0asε→0andEZε =EZ=1, then

lim

ε→0E|Zε−Z| =0.

For the proof see, e.g., Theorem A.4 in Ibragimov and Has’minskii (1981). Recall thatEϑ₀Zt,ε(v)=Eϑ0Zt(v)=1, where lnZt(v)=v∗t −1₂|v|2. Hence

for anyK >0

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Here we denoted|D|2_K = |D|2∧K. These allow us to write 1

(2M)d

KM

Eϑ0Zt,ε(v)Y¯t −u (t, Xt, ϑ0+εv)2dv

≥ 1

(2M)d

KM

Eϑ0Zt,ε(v)Y¯t−u (t, Xt, ϑ0+εv)2_Kdv

= 1

(2M)d

KM

Eϑ0Zt(v)u˙∗(v¯ε−v)2_Kdv (1+o(1)) .

Then

Zt(v)=exp

v∗t−

1 2|v|2

=exp

−1₂|v−t|2

exp

₁ 2|t|2

and

Eϑ₀Zt(v)u˙∗(v¯ε−v)2_K =Eϑ0e− 1

2|v−t|2_u˙∗_(v

ε−v)2_Ke

1 2|t|2

=Eϑ0e− 1

2|w|2_u˙∗₍_v¯_ε−_t −_w)2

Ke

1 2|¯t|2

=Eϑ₀e−12|w|2_u˙∗₍_w˜_ε−w)2

Ke

1 2|¯t|2

wherew=v−tandw˜ε = ¯vε−t. Introduce the setCMsuch that each coordinate

oft =

(t1), . . . , (d) t

is less thanM−√M, i.e.,(l)t ≤M− √

M. Then 1

(2M)dEϑ0e

1 2|t|2

w+t∈KM

u_˙∗(w˜ε−w)2_Ke−

1 2|w|2d_v

≥ 1

(2M)dEϑ0e

1 2|t|21

{t∈CM}

CM

u_˙∗(w˜ε−w)2_Ke−

1 2|w|2d_v

≥ 1

(2M)dEϑ0e

1 2|t|21

{t∈CM}

K√

M _u_˙∗₍_w_˜

ε−w)2_Ke−

1 2|w|2d_v

becauseK√

M ⊂CM. By Andersen’s Lemma (see, e.g., Ibragimov and Has’minskii

(1981), Lemma 2.10.2)

K√

M

u_˙∗(w˜ε−w)2_Ke−

1

2|w|2d_v≥

K√

M

u_˙∗w2_Ke−12|w|2d_v.

Note that asM→ ∞we obtain the limits 1

(2M)dEϑ0e

1 2|t|21

{t∈CM}−→

1 (2π )d2 and

1 (2π )d2

K√

M

u_˙∗w2_Ke−12|w|2d_v −→_Eϑ

0| ˙u∗t|2K.

The last steps areε→0 andK→ ∞

Eϑ₀u˙∗t2_K−→ ˙u◦(t, xt, ϑ0)∗It(ϑ0)−1u˙◦(t, xt, ϑ0) .

(13)

Therefore the bound (15) is verified. The bound (16) is proved in a similar way. Note thatZt =εux(t, Xt, ϑ) σ (t, Xt). An arbitrary estimatorZ¯t ofZt we write as

¯

Zt =εZ˜t. Then, forε−1Z¯t−Ztwe follow the proof given above.

DefinitionSuppose that the conditions L,R,U are fulfilled. Then we call the estimator-processesY_t∗, Z_t∗,0< t≤T asymptotically efficient if for allϑ0∈and

allt ∈(0, T]we have the equalities

lim

ν→0εlim→0_|_ϑ₋sup_ϑ₀_|≤_νε

−2_Eϑ_Y∗

t −Yt2= ˙u◦(t, xt, ϑ0)∗It(ϑ0)−1u˙◦(t, xt, ϑ0) , (17)

lim

ν→0εlim→0_|_ϑ₋sup_ϑ₀_|≤_νε

−4_Eϑ_Z∗

t −Zt2=(u˙◦)x(t, xt, ϑ0)∗It(ϑ0)−

1

2_{σ (t, x}_t₎2_.(18)

As we do not know the value ϑ we propose first to estimate it using some estimator-processϑε,t ,0< t≤T and then to put

Yt=u

t, Xt, ϑε

, Zt =ε

k

l=1

uxl

t, Xt, ϑε

σl(t, Xt) .

Recall that formally the MLE-processϑˆε,t,0< t≤T “solves” the problem and it

can be shown that under the supposed regularity conditions the estimator-processes

ˆ

Yt,ε =u(t, Xt,ϑˆε,t)andZˆt,ε =ux(t, Xt,ϑˆε,t)σ (t, Xt)are asymptotically efficient

in the sense of the relations (17) and (18), respectively, but this solution can not be called acceptable because the calculation ofϑˆε,tfor allt ∈(0, T], in the general case,

is a computationally difficult problem. That is why we propose to use the so-called multi-step MLE-process (Kutoyants 2015), which is introduced as follows. First we construct a preliminary estimatorϑ¯τεby the observationsX

τε =_(X

s,0≤s≤τε)on

some learning interval [0, τε], whereτε =εδwith 0< δ <1 and then we propose

an estimator-processϑ_t,ε , τε ≤ t ≤ T based on this preliminary estimator. Finally

we show that the corresponding estimators, say,Y_t,ε =ut, Xt, ϑt,ε

, τε ≤ t ≤ T

are asymptotically efficient.

As a preliminary we propose the minimum distance estimator (MDE)ϑ¯τε defined

by the relation

X− ˆXϑ¯τε 2

τε =

inf

ϑ∈

X− ˆX (ϑ)2

τε =

inf

ϑ∈

τε

0

Xt− ˆXt(ϑ )

2

dt.

Here the family of random processesXˆt(ϑ) ,0≤t ≤τε

, ϑ ∈

is defined as follows

ˆ

Xt(ϑ )=x0+

t

0 S (ϑ, s, Xs)ds, 0≤t≤τε, ϑ∈.

These estimators were studied in Kutoyants (1994) in the case of fixedτε=τ and

are called thetrajectory fitting estimators as well, because we choose an estimator

¯

ϑτε, which provides a trajectoryXˆt _¯

ϑτε

(14)

Xt,0≤t ≤τε. It was shown that if the conditions of regularity and the condition of

identifiability: for anyν >0

inf

ϑ0∈|ϑ−infϑ0|>ν

τ

0

₀t[S (ϑ, s, xs)−S (ϑ0, s, xs)] ds

2dt >0 (19)

hold and the matrix

Jτ(ϑ0)=

τ

0

t

0 ˙

S(ϑ0, s, xs)∗ds

t

0 ˙

S(ϑ0, s, xs)dsdt

is uniformly nondegenerate (belowλ∈Rd) inf

ϑ0∈|λinf|=1λ

∗_J

τ(ϑ0) λ >0, (20)

then the MDE is asymptotically normal

ε−1(ϑτ −ϑ0)=⇒Jτ(ϑ0)−1

τ

0

t

0 ˙

S(ϑ0, s, xs)∗ds

t

0 σ (s, xs)

∗_d_W

s dt.

Note that if we have the Regularity condition 3 (identifiability) withT =τ, then the identifiability condition (19) is also fulfilled. Indeed, suppose that there exists ϑ1=ϑ0such that

τ

0

₀t[S (ϑ1, s, xs)−S (ϑ0, s, xs)] ds

2dt=0. Then for allt ∈[0, τ]

t

0 S (ϑ1, s, xs)ds=

t

0 S (ϑ0, s, xs)ds,

which implies

S (ϑ1, s, xs)=S (ϑ0, s, xs) , 0≤s≤τ.

The last equality, of course, contradicts Regularity condition 3. Now suppose thatτε=εδwithδ <1 and the matrix

C(ϑ0)= ˙S(ϑ0,0, x0)∗S˙(ϑ0,0, x0) is uniformly nondegenerate inϑ0∈(belowλ∈Rd)

inf

ϑ∈_|λinf|=1λ

∗_C_(ϑ₀_{) λ >}₀ ₍₂₁₎

Then, we can obtain the asymptotics

ε−1(ϑτ −ϑ0)=

3 2√τεC

(ϑ0)−1σ (0, x0) 1

0

1−r2dW (r) (1+o(1)) .

Note that

Jτε(ϑ0) = τε

0 t 2_S_˙_(ϑ

0,0, x0)∗S˙(ϑ0,0, x0)dt (1+o(1))

= τε3

(15)

and

τε

0

t

0 S˙(ϑ0, s, xs)

∗_d_s t

0σ (s, xs)

∗_d_W

s dt

= ˙S(ϑ0,0, x0)∗σ (0, x0) τε

0 tWt dt (1+o(1))

= τ

5 2

ε

2 S˙(ϑ0,0, x0)∗σ (0, x0) 1

0

1−r2 dW (r) (1+o(1)) .

Therefore, the family of random vectorsε−1+δ2_ϑ¯_τ

ε−ϑ0

is asymptotically nor-mal. Moreover, following Kutoyants (1994) it can be shown that the moments are bounded, i.e.,

sup

ϑ0∈K

Eϑ0ε−1+

δ

2_ϑ¯_τ

ε−ϑ0 p

< C, (22)

where the constantC=C(p) >0 does not depend onεfor allp >0. Let us introduce the one-step MLE-processϑt,ε , τε≤t≤T

ϑt,ε = ¯ϑτε

+Itϑ¯τε −1 t

τε ˙

Sϑ¯τε, s, Xs ∗_A

(s, Xs)−1dXs−Sϑ¯τε, s, Xs

ds.(23)

Its properties are described in the following proposition.

Proposition 1Let the conditionsL,R be fulfilled and δ ∈ (0,1), then for all

t∈(0, T]

ε−1ϑ_t,ε −ϑ0⇒N

0,It(ϑ0)−1

and this estimator-process is asymptotically efficient. Moreover, we have the uniform consistency, i.e., for anyν >0

lim

ε_→0ϑsup0∈K

P(ε)_ϑ

0

sup

τε≤t≤T ϑ

t,ε−ϑ0> ν

=0.

ProofNote that the estimator ϑ

t,ε is defined for t ∈ [τε, T], but asτε → 0 we

obtain for any positivetthe relationt > τε.

The substitution of the observations (9) provides us the equality

ϑ_t,ε −ϑ0= ¯ϑτε−ϑ0+εIt _¯

ϑτε −1 t

τε ˙

Sϑ¯τε, s, Xs ∗

σ (s, Xs)−1dWs

+It

_¯ ϑτε

−1 t τε ˙

(s, Xs)−1

S (ϑ0, s, Xs)−S

_¯ ϑτε, s, Xs

ds.

Recall that the vector-process (Xs,0≤s≤T ) converges uniformly in s to the

deterministic vector-function(xs,0≤s≤T ) and the estimator ϑ¯τε is consistent.

(16)

It

_¯ ϑτε

−1 t τε ˙

Sϑ¯τε, s, Xs ∗

σ (s, Xs)−1dWs

−→It(ϑ0)−1

t

0 ˙

S(ϑ0, s, xs)∗σ (s, xs)−1dWs ∼N

0,It(ϑ0)−1

.

For the other terms, we first write the Taylor expansion

S (ϑ0, s, Xs)−Sϑ¯τε, s, Xs

= ˙S(ϑ0, s, Xs)∗ϑ0− ¯ϑτε

+O

ε2−δ

becauseϑ0− ¯ϑτε =O

ε1−δ2

. Then, we denote

Dε =It

_¯ ϑτε

− t τε ˙

(s, Xs)−1S˙(ϑ0, s, Xs)∗ds

and write

¯

ϑτε −ϑ0+It _¯

ϑτε −1 t

τε ˙

(s, Xs)−1S (ϑ0, s, Xs)−Sϑ¯τε, s, Xs _d

s

=ϑ¯τε−ϑ0

It

_¯ ϑτε

−1

Dε+O

ε2−δ

The following estimate can be easily verified

Dε =O

εδ+O

ε1−δ2

+O(ε)

becauseXs−xs =O(ε),ϑ¯τε −ϑ0=O

ε1−δ2

and

It(ϑ0)−

t

τε ˙

S(ϑ0, s, xs)∗A(s, xs)−1S˙(ϑ0, s, xs)ds=O

εδ.

Hence

ε−1ϑ_t,ε −ϑ0−It(ϑ0)−1 t

0 ˙

S(ϑ0, s, xs)∗σ (s, xs)−1dWs

=ε−1ϑ¯τε −ϑ0

Oε1−δ2

=Oε1−δ−→0. (24)

The uniform consistency can be shown following the proof of such uniform consistency presented in Kutoyants (2015), Theorem 1.

Let us define the estimator-processes Yε =

Yt,ε , τε≤t≤T

and Zε =

Zt,ε , τε≤t ≤Tas follows

Yt,ε =u

t, Xt, ϑt,ε

, Zt,ε =εux

t, Xt, ϑt,ε

σ (t, Xt) .

Theorem 2Suppose the conditions L,R,Uand(21)hold, then the esti-mator-processesYε, Zεadmit the representations

(17)

where the Gaussian process

ξt(ϑ0)=It(ϑ0)−1

t

0 ˙

S(ϑ0, s, xs)∗σ (s, xs)−1dWs, τε ≤t ≤T .

The random processes

ηt,ε =ε−1

Y_t,ε −Yt

, τ ≤t≤T ,

ζt,ε =ε−2

Zt,ε−Zt

, τ ≤t≤T

for anyτ ∈(0, T]converge in probability to the processes

ηt = ˙u◦(t, xt, ϑ0)∗ ξt(ϑ0) , τ ≤t ≤T ,

ζt = ˙u_◦,x(t, xt, ϑ0)∗ ξt(ϑ0) σ (t, xt) , τ ≤t≤T ,

respectively, uniformly int ∈[τ, T]. Moreover, these approximations are

asymptoti-cally efficient in the sense of (17), (18).

ProofBy the conditionU, we obtain the representation

Yt,ε −Yt =ut, Xt, ϑt,ε

−u (t, Xt, ϑ0)= ˙u(t, Xt, ϑ0)∗ϑt,ε −ϑ0

(1+o(1)) , Zt,ε −Zt =ε

ux

t, Xt, ϑt,ε

−ux(t, Xt, ϑ0)

σ (t, Xt) =εu˙_x(t, Xt, ϑ0)∗ϑt,ε −ϑ0

σ (t, Xt) (1+o(1)) ,

and for anyτ ∈(0, T]we have the convergence in probability

sup

τ≤t≤T| ˙

u(t, Xt, ϑ0)− ˙u◦(t, xt, ϑ0)| ≤ sup τ≤t≤T| ˙

u(t, Xt, ϑ0)− ˙u(t, xt, ϑ0)|

+ sup

τ≤t≤T| ˙

u(t, xt, ϑ0)− ˙u◦(t, xt, ϑ0)| −→0, (27)

sup

τ≤t≤T

u_˙x(t, Xt, ϑ0)− ˙ux(t, xt, ϑ0)≤ sup

τ≤t≤T

u_˙x(t, Xt, ϑ0)− ˙ux(t, xt, ϑ0)

+ sup

τ≤t≤T

u_˙_x(t, xt, ϑ0)− ˙u_◦,x(t, xt, ϑ0)−→0. (28)

Therefore, the representations (25),(26) follow now from (24).

More detailed analysis shows that the convergencesO(1)in (24),(25) are uniform int ∈[τ, T] due to (11). Moreover, we have the convergence of moments uniform on compactsϑ0∈Kas well, because we have (12) and the moments of the preliminary

estimator are bounded (22). Therefore, the estimates used above can be also written for the moments. This convergence of moments provides the asymptotic efficiency of the estimatorsYε, Zε.

The estimatorsYt,ε , Zt,ε, τε ≤t ≤ T are given for the valuest > τε =εδ with

(18)

which uses the preliminary estimator with the worse rate of convergence. Let us take δ∈ [1,4₃), introduce the second preliminary estimator-process

¯

ϑt,ε= ¯ϑτε+It _¯

ϑτε −1 t

τε ˙

(s, Xs)−1

dXs−S

_¯ ϑτε, s, Xs

ds

and the two-step MLE-processϑt,ε, τε≤t ≤T

ϑt,ε = ¯ϑt,ε+Itϑ¯t,ε−1_τt εS˙

_¯ ϑτε, s, Xs

∗_A

(s, Xs)−1dXs−Sϑ¯t,ε, s, Xsds.

For the preliminary estimator we obtain the same estimate (22), but with different τε. Further, for the first preliminary estimator similar calculations as above provide

us the estimates

ε−γϑ¯t,ε−ϑ0=ε−γϑ¯τε −ϑ0 2

O(1)+o(1)=ε−γ+2−δO(1)+o(1). For the two-step MLE-process we have

ε−1ϑt,ε−ϑ0

=εγ−δ2_ε−γ_ϑ¯_t,ε−_ϑ₀ _(ε−1+δ2_ϑ¯_τ

ε −ϑ0)O(1) +It(ϑ0)−1

t

τε ˙

S(ϑ0, s, xs)∗A(s, xs)−1dWs+o(1).

Therefore if we takeγ such thatγ +δ <2 andγ −δ₂ >0, say,γ < 2₃, then we obtain

ε−1ϑt,ε−ϑ0

=⇒N0,It(ϑ0)−1

.

Now the estimator-processesYε, Zε defined with the help of two-step

MLE-process

Yt,ε = u

t, Xt, ϑt,ε

, τε≤t≤T ,

Zt,ε = ux

t, Xt, ϑt,ε

σ (t, Xt) , τε ≤t ≤T

are known for the larger time interval [τε, T].

Of course, we can continue this process and to reduce the learning interval once more by introducing the three-step MLE-processϑt,εas follows. The learning

inter-val is [0, τε],τε = εδ, whereδ ∈ [4₃,3₂). The first preliminary estimator-process

is

¯

ϑt,ε = ¯ϑτε+It _¯

ϑτε −1t

τεS˙ _¯

ϑτε, s, Xs ∗_A

(s, Xs)−1dXs−Sϑ¯τε, s, Xs

ds, the second is

¯¯

ϑt,ε = ¯ϑt,ε+Itϑ¯t,ε−1_τt εS˙

_¯ ϑτε, s, Xs

∗_A

(s, Xs)−1dXs−Sϑ¯t,ε, s, Xsds,

and the three-step MLE-process

ϑt,ε = ¯¯ϑt,ε+It(ϑ¯¯t,ε)−1_τt εS˙

_¯ ϑτε, s, Xs

∗_A

(s, Xs)−1

dXs−S(ϑ¯¯t,ε, s, Xs)ds

.

The similar calculations will provide us the relations :ϑ¯τε−ϑ0=ε1− δ

2_O(1₎,

ε−γ1_ϑ¯

t,ε−ϑ0=ε2−γ1−δO(1), ε−γ2

¯¯

ϑt,ε−ϑ0

(19)

and

ε−1ϑt,ε−ϑ0

=εγ2−δ2

ε−γ2_ϑ¯¯

t,ε−ϑ0

(ε−1+δ2_ϑ¯_τ

ε−ϑ0)O(1) +It(ϑ0)−1

t

τε ˙

S(ϑ0, s, xs)∗A(s, xs)−1dWs +o(1).

Hence if we choseδ,γ1andγ2such that

δ <2, γ1+δ <2, γ1−γ2+1− δ

2 >0, γ2> δ 2, then once more we obtain asymptotically efficient MLE-process

ε−1ϑt,ε−ϑ0

=⇒N0,It(ϑ0)−1

.

Therefore, we obtain the corresponding approximationsY

t,ε , Zt,ε for the values

t∈[τε, T] with essentially smallerτεthan in the case of one-step MLE-process.

Example

Black and Scholes model.Suppose that the forward equation is

dXt =ϑ Xtdt+εσ XtdWt, X0=x0>0, 0≤t ≤T ,

and the functionsf (x, y, z)= −βy−γ xzand(x)are given. The function(x) is continuous and satisfies the condition|(x)| ≤C1+ |x|pwith some constants C >0 andp >0. We have to find(Yt, Zt)such that

dYt =[βYt+γ XtZt] dt+ZtdWt, 0≤t ≤T ,

andYT = (XT).

The corresponding PDE is

∂u

∂t +

ε2σ2x2 2

∂2u

∂x2 +(ϑ−εσ γ )x ∂u

∂x−βu=0, u(T , x, ϑ )=(x).

To write its solution we change the variables s = T −t,x¯ = lnx and let u (t,x, ϑ )¯ =eμ(ϑ)x¯+λ(ϑ)sv (s,x, ϑ )¯ , where

μ(ϑ)=2εσ γ+εσ

2₋₂_ϑ

2ε2σ2 , λ(ϑ )=β+

2εσ γ+εσ2−2ϑ2 8ε2σ2 . Then, we obtain the reduced equation

∂v

∂s =

ε2σ2 2

∂2v

∂x¯2, 0≤s≤T , v(0,x, ϑ )¯ =e

−μ(ϑ)x¯_(ex¯_),

whose solution is well-known

v(s,x, ϑ )¯ =√ 1 2π sε2σ2

_∞

−∞exp

−(₂x¯−z)2

ε2σ2s

(20)

Let us fixτε =ε

3

4and introduce the preliminary TFE is

¯

ϑτε = τε

0 (Xs−x0) Ht dt

τε 0 Ht2dt

, Ht =

t

0Xs ds.

Of course, we also can write the MLEϑˆτε

ˆ ϑτε = 1 τε τε 0

dXs

Xs

,

but as in our work we used the TFE, we show how to calculateϑ¯τε. The Fisher

information is_It(ϑ)=tσ−2. The one-step MLE-process is

ϑ_t,ε = ¯ϑτε+

1 t t τε 1 Xs

dXs− ¯ϑτεXsds

.

Moreover, it is easy to see that

ϑ_t,ε = ¯ϑτε+

1 t

t

τε

dXs

Xs − ¯

ϑτε t−τε

t =

1 t

t

τε

dXs

Xs + ¯

ϑτε τε

t

= ˆϑt,ε+ ¯ϑτε τε t − 1 t τε 0

dXs

Xs = ˆ

ϑt,ε+o(ε).

Hence, the estimatorsϑt,ε andϑˆt,ε have the same limit distributions. Therefore

the estimator-process

Yt,ε =X μ(ϑ¯_τε) t

eλ(ϑ¯τε)(T−t )

2π (T −t) ε2σ2

_∞

−∞e

−(lnXt−z)2

2ε2σ2(T−t)−μ(ϑ¯τε)z_ezd_z.

It is easy to see thatYt,ε −→ (XT)ast →T. The expression forZt,εcan be

written as well.

Discussions

Note that we approximate the solution of the BSDE and not the equation itself. Of course, it is also possible to write the stochastic differential forYt,ε . For

simplic-ity of notation we consider the case k = 1, d = 1. Indeed, the process Y_t,ε = ut, Xt, ϑt,ε

, whereXt has stochastic differential (9) andϑt,ε given by (23) can be

written as follows

ϑt,ε = ¯ϑτε+εIt _¯

ϑτε −1 t

τε ˙

S(ϑ¯τε, s, Xs)σ (s, Xs)− 1_d_W

s

+Itϑ¯τε −1 t

τε ˙

S(ϑ¯τε, s, Xs)A(s, Xs)− 1

S(ϑ0, s, Xs)−S(ϑ¯τε, s, Xs) _d

s

= ¯ϑτε+εI− 1 t

t

τε

αs,εdWs+It−1

t

τε

βs,εds (29)

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It was shown that the right-hand side of (29) tends to a constantϑ0asε→0 and we can verify that dϑt,ε →0.

More detailed analysis shows that

dY_t,ε = −f (t, Xt, Yt, Zt)dt+ZtdWt+εdηt +o(ε), τε≤t≤T ,

where the Gaussian processηt is defined in Theorem 2. We used the relationYt,ε =

Yt+εηt +o(ε).

The multi-step MLE-processes used in this work can be useful in similar problems of BSDE approximations for dicrete-time observations and ergodic diffusion models mentioned in the introduction (see (Abakirova, A and Kutoyants, YA: On approxi-mation of the BSDE. Large samples approach.In preparation) and Gasparyan and Kutoyants (2015)).

Acknowledgements This work was done with partial financial support of the RSF grant number 14-49-10079.

Competing interests

I declare that there is no competing interests.

Authors’ contributions

I read and approved the final manuscript.

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